Optimal. Leaf size=82 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (b x+2)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]
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Rubi [A] time = 0.0533578, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (b x+2)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(2 + b*x)^(3/2),x]
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Rubi in Sympy [A] time = 8.75426, size = 73, normalized size = 0.89 \[ \frac{\sqrt{x} \left (b x + 2\right )^{\frac{5}{2}}}{3 b} - \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{6 b} - \frac{\sqrt{x} \sqrt{b x + 2}}{2 b} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+2)**(3/2)*x**(1/2),x)
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Mathematica [A] time = 0.0633419, size = 60, normalized size = 0.73 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (2 b^2 x^2+7 b x+3\right )}{6 b}-\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(2 + b*x)^(3/2),x]
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Maple [A] time = 0.007, size = 87, normalized size = 1.1 \[{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}+{\frac{1}{2\,b}\sqrt{x}\sqrt{bx+2}}-{\frac{1}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+2)^(3/2)*x^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(3/2)*sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.219929, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b^{2} x^{2} + 7 \, b x + 3\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 3 \, \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{6 \, b^{\frac{3}{2}}}, \frac{{\left (2 \, b^{2} x^{2} + 7 \, b x + 3\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} - 6 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{6 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(3/2)*sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 21.919, size = 92, normalized size = 1.12 \[ \frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{b x + 2}} + \frac{11 b x^{\frac{5}{2}}}{6 \sqrt{b x + 2}} + \frac{17 x^{\frac{3}{2}}}{6 \sqrt{b x + 2}} + \frac{\sqrt{x}}{b \sqrt{b x + 2}} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+2)**(3/2)*x**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(3/2)*sqrt(x),x, algorithm="giac")
[Out]